Activity 4.6 Mathematical Modeling Answers

Mathematical Model

Mathematical models (formulae) are available (Pacejka, 2012) for use in CAE simulation and prediction of vehicle braking and cornering.

From: Braking of Road Vehicles , 2014

Mathematical models

Huw Fox , Bill Bolton , in Mathematics for Engineers and Technologists, 2002

3.2 Relating models and data

In testing mathematical models against real data, we often have the situation of having to check whether data fits an equation. If the relationship is linear, i.e. of the form y = mx + c, then it is comparatively easy to see whether the data fits the straight line and to ascertain the gradient m and intercept c. However, if the relationship is non-linear this is not so easy. A technique which can be used is to turn the non-linear equation into a linear one by changing the variables. Thus, if we have a relationship of the form y = ax ² + b, instead of plotting y against x to give a non-linear graph we can plot y against x ² to give a linear graph with gradient a and intercept b. If we have a relationship of the form y = a/x we can plot a graph of y against 1/x to give a linear graph with a gradient of a.

Example

The following data was obtained from measurements of the load lifted by a machine and the effort expended. Determine if the relationship between the effort E and the load W is linear and if so the relationship.

E in N	18	27	32	43	51
W in N	40	80	120	160	200

Within the limits of experimental error the results appear to indicate a straight-line relationship (Figure 3.14). The gradient is 41/200 or about 0.21. The intercept with the E axis is at 10. Thus the relationship is E = 0.21 W + 10.

Figure 3.14. Example

Example

It is believed that the relationship between y and x for the following data is of the form y = ax ² + b. Determine the values of a and b.

Figure 3.15 shows the graph of y against x ². The graph has a gradient of AB/BC = 12.5/25 = 0.5 and an intercept with the y-axis of 2. Thus the relationship is y = 0.5x ² + 2.

Figure 3.15. Example

Problems 3.2

1

Determine, assuming linear, the relationships between the following variables:

(a)

The load L lifted by a machine for the effort E applied.

E in N	9.5	11.8	14.1	16.3	18.5
L in N	10	15	20	25	30

(b)

The resistance R of a wire for different lengths L of that wire.

R in W	2.1	4.3	6.3	8.3	10.5
L in m	0.5	1.0	1.5	2.0	2.5

2

Determine what form the variables in the following equations should take when plotted in order to give straight-line graphs and what the values of the gradient and intercept will have.

(a)

The period of oscillation T of a pendulum is related to the length L of the pendulum by the equation:

$T=2\pi \sqrt{\frac{L}{g}}$

where g is a constant.

(b)

The distance s travelled by a uniformly accelerating object after a time t is given by the equation:

$s=ut+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.a{t}^2$

where u and a are constants.

(c)

The e.m.f. e generated by a thermocouple at a temperature θ is given by the equation;

$e=a\theta +b{\theta }^2$

where a and b are constants.

(d)

The resistance R of a resistor at a temperature h is given by the equation:

$R=R_0+R_0\alpha \theta$

where R ₀ and α are constants.

(f)

The pressure p of a gas and its volume V are related by the equation:

$pV=k$

where k is a constant.

(g)

The deflection y of the free end of a cantilever due to it own weight of w per unit length is related to its length L by the equation:

$y={\frac{wL}{8EI}}^4$

where w, E and I are constants.

3

The resistance R of a lamp is measured at a number of voltages V and the following data obtained. Show that the law relating the resistance to the voltage is of the form R = (a/V) + b and determine the values of a and b.

R in Ω	70	62	59	56	55
V in V	60	100	140	200	240

4

The resistance R of wires of a particular material are measured for a range of wire diameters d and the following results obtained. Show that the relationship is of the form R = (a/d ²) + b and determine the values of a and b.

R in Ω	0.25	0.16	0.10	0.06	0.04
d in mm	0.80	1.00	1.25	1.60	2.00

5

The volume V of a gas is measured at a number of pressures p and the following results obtained. Show that the relationship is of the form V = ap^b and determine the values of a and b.

V in m3	13.3	11.4	10.0	8.9	8.0
p in l05Pa	1.2	1.4	1.6	1.8	2.0

6

When a gas is compressed adiabatically the pressure p and temperature T are measured and the following results obtained. Show that the relationship is of the form T = aρ ^b and determine the values of a and b.

p in 105 Pa	1.2	1.5	1.8	2.1	2.4
T in K	526	560	589	615	639

7

The cost C per hour of operating a machine depends on the number of items n produced per hour. The following data has been obtained and is anticipated to follow a relationship of the form C = an ³ + b. Show that this is the case and determine the values of a and b.

C in £	31	38	67	94	155
n	10	20	30	40	50

8

The following are suggested braking distances s for cars travelling at different speeds v. The relationship between s and v is thought to be of the form s = av ² + bv. Show that this is so and determine the values of a and b.

s in m	5	15	30	50	75
v in m/s	5	10	15	20	25

Hint: consider s/v as one of the variables.

9

The luminosity I of a lamp depends on the voltage V applied to it. The relationship between I and V is thought to be of the form I = aV ^b . Use the following results to show that this is the case and determine the values of a and b.

I in candela	3.6	6.4	10.0	14.4	19.6
V in volts	60	80	100	120	140

10

From a lab test, it is believed that the law relating the voltage v across an inductor and the time t is given by the relationship v = A e ^t/B , where A and B are constant and e is the exponential function. From the lab test the results observed were:

v (volts)	908.4	394.8	171.6	32.4	14.1	6.12
t (ms)	10	20	30	50	60	70

Show that the law relating the voltage to time is, in fact, true. Then determine the values of the constants A and B.

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FUEL CELLS – MOLTEN CARBONATE FUEL CELLS | Modeling

Z. Ma , ... M. Farooque , in Encyclopedia of Electrochemical Power Sources, 2009

Conclusions

Mathematical models are effective tools for MCFC cell and stack design and optimization. Because the fuel cell process involves transport processes and electrochemical reactions, which are interrelated and affect the fuel cell performance, mathematical models provide the insight and understanding in guiding the design and development. Evolution of MCFC models from electrode, cell, and stack has proceeded in parallel with cell and stack technology development. Mathematical models have been used successfully in understanding fundamental processes at the electrodes and also to design cell and stack components. The model-based design has considerably reduced the design cycle time and minimized development risks and resulted in a stack design with a 20% higher output and reduced temperature differential, leading to lower power system cost.

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Methods to develop mathematical models: traditional statistical analysis

Jorge Garza-Ulloa , in Applied Biomechatronics using Mathematical Models, 2018

Abstract

Mathematical models for kinematics, kinetics, and muscles potentials activities from sEMG based on traditional statistical analysis are developed using different methods for data analysis, where each model is represented using a structure with a linear dynamic form, explicit and discrete, that can be verified as stochastic process and arising from empirical finding. In this chapter, Mathematical tools are studied with the objective of obtaining Mathematical Models from: traditional stochastic methods from probability and statistics as probability models, probability distributions, statistical inferences using statistical hypotheses testing parameters, z-tests, t-tests, paired t-tests, ANOVA. We apply them to: Linear equations, Regression methods, and Autoregressive equations. The different methods explained are applied to research Biomechanics examples to model and detect data behaviors, and this chapter is concluded with the development of a special software application of Mathematical Models for Analysis of Continuous Glucose Monitor (CGM) for Diabetic subjects. Note: Others Mathematical Models based on Domain/Conversion/Transform analysis, and Machine Learning Models Analysis are studied in the next chapters.

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Theory of Vibration

DAN B. MARGHITU , ... DUMITRU MAZILU , in Mechanical Engineer's Handbook, 2001

3.3 System Model

3.3.1 VIBRATIONS WITH ONE DEGREE OF FREEDOM

Mathematical models from Eqs. (3.24) and (3.25) are open monovariable systems (with one input and one output). The input value is i(t) = F(t) or, respectively, i(t) = ϕ(t). The output value is e(t) = q(t). The mathematical model can be written as

(3.28) $\begin{array}{l}\dot{x}=[A]x+[B]1\\ e=[C]x,\end{array}$

where

$[A]=\left[\begin{array}{ll}0\hfill & 1\hfill \\ -{\omega }_n^2\hfill & -2n\hfill \end{array}\right],[B]=\left[\begin{array}{l}0\hfill \\ 1\hfill \end{array}\right],[C]=\left[\begin{array}{ll}1\hfill & 0\hfill \end{array}\right],i=\varphi (t),e=q(t).$

3.3.2 VIBRATION WITH A FINITE NUMBER OF DEGREES OF FREEDOM

The mathematical model of Eqs. (3.26) and (3.27) are also open monovariable linear systems. The input vector is i = [F], the output vector is e = [q] and the state vector is

$x=\left[\begin{array}{l}[q]\hfill \\ [\dot{q}]\hfill \end{array}\right].$

From the matrix form of Eq. (3.27), the canonical form results:

(3.29) $\begin{array}{c}\dot{x}=[A]x+[B]\varphi \\ [q]=[C]x.\end{array}$

Here

$\begin{array}{ll}\hfill {[A]}_{2n\times 2n}& =\left[\begin{array}{cc}{[0]}_{n\times n}\hfill & {[I]}_{n\times n}\hfill \\ -{[M]}^{-1}[R]\hfill & -{[M]}^{-1}[D]\hfill \end{array}\right],{[B]}_{2n\times 2n}=\left[\begin{array}{ll}{[0]}_{n\times n}\hfill & {[0]}_{n\times n}\hfill \\ {[I]}_{n\times n}\hfill & {[0]}_{n\times n}\hfill \end{array}\right]\hfill \\ \hfill \varphi & =[\varphi ]=\left[\begin{array}{l}{[M]}^1[F]\hfill \\ {[0]}_{n\times 1}\hfill \end{array}\right],[C]=\left[\begin{array}{ll}{[I]}_{n\times n}\hfill & {[0]}_{n\times n}\hfill \end{array}\right].\hfill \end{array}$

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From mathematical modeling and machine learning to clinical reality

Ben D. MacArthur , ... Richard O.C. Oreffo , in Principles of Tissue Engineering (Fifth Edition), 2020

Abstract

Mathematical models are routinely used in the physical and engineering sciences to help understand complex systems and optimize industrial processes. There are numerous examples of the fruitful application of mathematical principles to problems in cell and molecular biology, and recent years have seen increasing interest in applying quantitative techniques to problems in biotechnology. This chapter reviews some ways, in which mathematical models and machine learning may be used to advance our understanding of the complex biological processes involved in cellular differentiation and tissue growth and development, for applications in tissue engineering and regenerative medicine. We will discuss how mathematical models can advance our understanding of stem cell differentiation; growth and development of stem and progenitor cell colonies; and the mechanisms that underpin spatial organization of structure in three-dimensional developing tissues. We will also discuss how recent developments in machine leaning are able to extract biological knowledge from the most complex experimental datasets. We will conclude by considering how computational techniques can be further applied to the design and optimization of effective tissue engineering strategies for clinical application. Although this field is in its infancy, the appropriate use of mathematical methods has considerable potential to transform tissue engineering from experimental concept to clinical reality.

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Virtual CNC machine tool modeling and machining simulation in high speed milling

Anthony Chukwujekwu Okafor , in High Speed Machining, 2020

5.1 Conclusion

Mathematical model representation of WEBNHE geometry and its mechanistic cutting force model incorporating the effects of emulsion and MQL cooling strategies were developed and used for predicting cutting forces in the high speed end milling of Inconel 718. The following conclusions can be drawn:

1.

Mechanistic cutting force model for WEBNHE incorporating the effects of cutting force and edge force coefficients, and cooling strategies (conventional emulsion and MQL cooling) was successfully developed and experimentally validated for predicting cutting forces in end milling Inconel-718.

2.

WEBNHE geometry is represented by a mathematical model, using polar coordinate and cubic spline approximation to represent the geometries of the envelope and the wavy-cutting edges, respectively.

3.

MATLAB codes were developed and used to simulate mathematical model of WEBNHE geometry and to simulate cutting forces in high speed end milling of Inconel 718.

4.

Predicted cutting force components, F _x , F _y , and F _z , were very much in agreement with measured values in both shape and magnitude.

5.

High spindle speed of 93 rpm of WEBNHE generates lower cutting forces than low spindle speed of 62 rpm.

6.

MQL generates lower cutting force than emulsion cooling.

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Application of mathematical models in biomechatronics: artificial intelligence and time-frequency analysis

Jorge Garza-Ulloa , in Applied Biomechatronics using Mathematical Models, 2018

Abstract

Mathematical models for kinematics, kinetics, and muscles potentials activities are deducted of data signals analysis, using time-frequency domain and non-classic methods from pattern recognitions to computational learning theory of Artificial Intelligence (AI) based on Machine Learning algorithms. Covering decision theory for supervised, and unsupervised learning as: Partitional Clustering (k-means algorithm), Hierarchical Clustering, Artificial Neural Network (ANN), and others approaches. Applying them in practical Biomedical research examples for Classification, and Regression using: Classification Tree Analysis (CTA), and Regression Tree Analysis (RTA). Finalizing, with Mathematical models of Soft Computing for Fuzzy Inference Systems (FIS), Fuzzy Relations and Fuzzy Similarities deducted from the dynamics of human body, using synchronized signals from: Ground Reaction Forces (GRF) during the gait phases, muscles activities of Surface Electromyography (SEMG), and Joint Angles.

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Microscale and Macroscale Modeling of Microalgae Cultivation in Photobioreactor: A Review and Perspective

Choon Gek Khoo , ... Keat Teong Lee , in Advances in Feedstock Conversion Technologies for Alternative Fuels and Bioproducts, 2019

1.3 Mathematical Models

Mathematical models are one of the important interpretations of experimental results that provide a greater insight toward the studied biological system [10]. Throughout the decades, there has been a great deal of importance placed on predicting the productivity of the microalgae biomass and the transport phenomena in PBR through mathematic modeling [11]. The mathematical models can be conceptually categorized into macroscale and microscale, which described the operational performance in PBRs and the growth of microalgae cells, respectively [12–14]. Fig. 1.1 illustrates the phenomena occurring at the macroscale and microscale of microalgae growth in PBR. Generally, the developed mathematical modeling follows the sequences from microscale of microalgae growth to the macroscale of transport phenomena in PBR. The integration between micro- and macroscale modeling are theoretically driven, which is applicable to the scale-up the microalgae cultivation process. However, the complexity of macroscale modeling with respect to the simple expressions of microscale modeling is not widely reported.

Figure 1.1. Phenomena occurring during microalgae cultivation within photobioreactors: (A) macroscale transport phenomena and (B) microscale kinetic growth for microalgae cells.

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Direct Circulation Models

William C. Lyons , ... Tom Weller , in Air and Gas Drilling Manual (Fourth Edition), 2021

Abstract

Mathematical models for the flow of compressed air (or other gases) through the direct circulation drill string with flow down the inside of the drill string and return flow up the annulus with entrained rock cuttings are presented and discussed. These compressible air flow models include separate models for the basic two-phase air drilling with entrained cuttings model, the three-phase aerated fluid drilling model (air, incompressible fluid, and entrained cuttings), and the stable foam drilling models. The basic two-phase flow model is derived in this chapter.

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Activity 4.6 Mathematical Modeling Answers

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